The document discusses correlation coefficients (both simple and partial) and multiple correlation, detailing how they measure the degree of linear association between variables. It outlines hypothesis testing for these coefficients, providing examples and calculations to illustrate the concepts. Additionally, it explains the importance of controlling for other variables to assess true relationships among factors.

1. Simple, Partial and Multiple
Correlation

2. Scatter Plot Examples
y
x
y
x
y
y
x
x
Strong relationships Weak relationships
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3. Scatter Plot Examples
y
x
y
x
No relationship
(continued)
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4. r = +0.3 r = +1
Examples of Approximate “r” values
y
x
y
x
y
x
y
x
y
x
r = -1 r = -0.6 r = 0
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5. Correlation Coefficient
• Measure of the degree of linear association between two variables.
• The population correlation coefficient ρ (rho)
• The sample correlation coefficient r is an estimate of ρ
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6. Features of ρ and r
•Range between -1 and 1
•It is symmetrical w.r.t variables i.e., rxy = ryx
•Unit free
•“r” is independent of change in origin and scale
• The closer to -1, the stronger the negative linear relationship
• The closer to 1, the stronger the positive linear relationship
• The closer to 0, the weaker the linear relationship
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7. • The value of r ranges between ( -1) and ( +1)
• The value of r denotes the strength of the association as
illustrated by the following diagram.
-1 1
0
-0.25
-0.75 0.75
0.25
strong strong
intermediate intermediate
weak weak
no relation
perfect
correlation
perfect
correlation
Direct
indirect
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8. Example
• The following data represent the Study hours and
the Marks of 7 students in the subject of
Mathematics. Find the Correlation Coefficient ‘r’
and interpret it.
• Coefficient of Correlation r = 0.804
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Study hours
X
Marks
Y
5 50
3 35
5 45
3 26
4 30
3 35
4 40
𝑟 =
𝑥𝑦
𝑥2 𝑦2

9. Example
• The following data represent the
wing length and tail length of
sparrows
Wing length
(X)
Tail length (Y)
10.4 7.4
10.8 7.6
11.1 7.9
10.2 7.2
10.3 7.4
10.2 7.1
10.7 7.4
10.5 7.2
10.8 7.8
11.2 7.7
10.6 7.8
11.4 8.3
128.2 90.8
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𝑟 =
𝑥𝑦
𝑥2 𝑦2

10. Hypothesis Testing about Correlation Coefficient
• Step-I: Formulation of hypothesis
• Step-II: Level of Significance: α=0.05
• Step-III: Test Statistics
• Step-IV: Calculations:
• Step-V: Decision Rule:
• Step-VI: Conclusion
𝐻0: 𝜌 = 0, 𝜌 ≥ 0, 𝜌 ≤ 0
𝐻1: 𝜌 ≠ 0, 𝜌 < 0, 𝜌 > 0
Case-I:- Population correlation co-efficient ρ is equal to zero
𝑡 =
𝑟 − 𝜌
𝑆𝐸 𝑟
𝑤ℎ𝑒𝑟𝑒, 𝑆𝐸 𝑟 =
1 − 𝑟2
𝑛 − 2
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Hypothesis Decision Rules
𝐻1: 𝜌 ≠ 0 𝑡𝑐𝑎𝑙 ≤ −𝑡𝛼
2
, 𝑛−2 OR
𝑡𝑐𝑎𝑙 ≥ 𝑡𝛼
2
, 𝑛−2
𝐻1: 𝜌 > 0 𝑡𝑐𝑎𝑙 ≥ 𝑡𝛼,(𝑛−2)
𝐻1: 𝜌 < 0 𝑡𝑐𝑎𝑙 ≤ −𝑡𝛼 ,(𝑛−2)

11. Hypothesis Testing about Correlation Coefficient
• Step-I: Formulation of hypothesis
• Step-II: Level of Significance: α=0.05
• Step-III: Test Statistics
• Step-IV: Calculations:
• Step-V: Decision Rule:
• Step-VI: Conclusion
Case-2:- Population correlation co-efficient ρ is equal to ρ0 where ρ0 not equal to zero.
𝐻0: 𝜌 = 𝜌0, 𝜌 ≥ 𝜌0, 𝜌 ≤ 𝜌0
𝐻1: 𝜌 ≠ 𝜌0, 𝜌 < 𝜌0, 𝜌 > 𝜌0
𝑍 =
𝑍𝑓 − 𝜇𝑧
𝑆𝐸 𝑍𝑓
, 𝑤ℎ𝑒𝑟𝑒
𝑍𝑓 =
1
2
𝑙𝑛
1 + 𝑟
1 − 𝑟
, 𝜇𝑧 =
1
2
𝑙𝑛
1 + 𝜌
1 − 𝜌
𝑆𝐸 𝑍𝑓 =
1
𝑛 − 3
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Hypothesis Decision Rules
𝐻1: 𝜌 ≠ 𝜌0 𝑍𝑐𝑎𝑙 ≤ −𝑍𝛼
2
OR
𝑍𝑐𝑎𝑙 ≥ 𝑍𝛼
2
𝐻1: 𝜌 > 𝜌0 𝑍𝑐𝑎𝑙𝑐 ≥ 𝑍𝛼
𝐻1: 𝜌 < 𝜌0 𝑍𝑐𝑎𝑙𝑐 ≤ −𝑍𝛼

12. Example
A random sample of size 28 pairs from a bivariate normal population
showed a correlation coefficient of 0.7. Is this value consistent with the
assumption that the correlation coefficient in the population is 0.5?
• Step 1: Hypothesis
• Step 2: Choose α
• Step-3: Test Statistics
Z = 1.6
• Step-4: Calculations
• Step-5: Decision Rule
• Step-6: Conclusion
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13. Partial Correlation
• The relationship between two variables may be affected by other variables
which either strengthen or weakens the relationship.
• Partial correlation is a measure of the strength of a relationship between two
variables while controlling for the effect of one or more other variables.
• Monthly Income and Education Level of an individual is affected by the
Experience of the individual. To get the REAL relationship between two
variables other extraneous factors which are suspected to affect the
relationship are controlled or partial out by the use of partial correlation
coefficients.
• Similarly, you might want to see if there is a correlation between caloric intake
and blood pressure, while controlling for weight or amount of exercise.
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14. Partial Correlation Coefficient
• If we have three variables X1, X2, and X3 then the population partial correlation
coefficient between X1 and X2, keeping the effect of X3 constant is denoted by
ρ12.3(read as rho one two dot three) and can be calculated in terms of simple
correlation coefficients as follows.
• r12 is the simple correlation coefficient between X1 and X2
• r13 is the simple correlation coefficient between X1 and X3
• r23 is the simple correlation coefficient between X2 and X3
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𝑟12.3 =
𝑟12 − 𝑟13𝑟23
1 − 𝑟13
2
∗ 1 − 𝑟23
2
r12 = r21
r13 = r31
r23 = r32

15. r13.2 and r23.1
• Similarly we can compute
• r12.3 r13.2 and r23.1 are known as first order partial correlation.
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𝑟13.2 =
𝑟13 − 𝑟12𝑟23
1 − 𝑟12
2
∗ 1 − 𝑟23
2
𝑟23.1 =
𝑟23 − 𝑟12𝑟13
1 − 𝑟12
2
∗ 1 − 𝑟13
2

16. Testing of hypothesis for Partial Correlation
The procedure of testing of hypothesis for Partial Correlation is similar to the
Simple Correlation
Case-I:- Population correlation co-efficient ρ12.3 is equal to zero
Case-2:- Population correlation co-efficient ρ12.3 is equal to some value other
than zero
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17. Testing of hypothesis for Partial Correlation
• Step-I: Formulation of hypothesis
• Step-II: Level of Significance: α=0.05
• Step-III: Test Statistics
• Step-IV: Calculations:
• Step-V: Decision Rule:
• Step-VI: Conclusion
Case-I:- Population correlation co-efficient ρ12.3 is equal to zero
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𝐻0: 𝜌12.3 = 0, 𝜌12.3 ≥ 0, 𝜌12.3 ≤ 0
𝐻1: 𝜌12.3 ≠ 0, 𝜌12.3 < 0, 𝜌12.3 > 0
Hypothesis Decision Rules
𝐻1: 𝜌12.3 ≠ 0 𝑡𝑐𝑎𝑙 ≤ −𝑡𝛼
2
, 𝑛−𝑞−2 OR
𝑡𝑐𝑎𝑙 ≥ 𝑡𝛼
2
, 𝑛−𝑞−2
𝐻1: 𝜌12.3 > 0 𝑡𝑐𝑎𝑙𝑐 ≥ 𝑡𝛼,(𝑛−𝑞−2)
𝐻1: 𝜌12.3 < 0 𝑡𝑐𝑎𝑙𝑐 ≤ −𝑡𝛼 ,(𝑛−𝑞−2)
𝑡 =
𝑟12.3 − 𝜌12.3
𝑆𝐸 𝑟12.3
𝑤ℎ𝑒𝑟𝑒, 𝑆𝐸 𝑟12.3 =
1 − 𝑟12.3
2
𝑛 − q − 2
, q is the number of variable kept constant

18. Testing of hypothesis for Partial Correlation
• Step-I: Formulation of hypothesis
• Step-II: Level of Significance: α=0.05
• Step-III: Test Statistics
• Step-IV: Calculations:
• Step-V: Decision Rule:
• Step-VI: Conclusion
Case-II: Population correlation co-efficient ρ12.3 is equal to some value other than zero
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𝐻0: 𝜌12.3 = 𝜌0, 𝜌12.3 ≥ 𝜌0, 𝜌12.3 ≤ 𝜌0
𝐻1: 𝜌12.3 ≠ 𝜌0, 𝜌12.3 < 𝜌0, 𝜌12.3 > 𝜌0
𝑍 =
𝑍𝑓 − 𝜇𝑧
𝑆𝐸 𝑍𝑓
, 𝑤ℎ𝑒𝑟𝑒
𝑍𝑓 =
1
2
𝑙𝑛
1 + 𝑟12.3
1 − 𝑟12.3
, 𝜇𝑧 =
1
2
𝑙𝑛
1 + 𝜌12.3
1 − 𝜌12.3
𝑆𝐸 𝑍𝑓 =
1
𝑛 − 𝑞 − 3
Hypothesis Decision Rules
𝐻1: 𝜌12.3 ≠ 𝜌0 𝑍𝑐𝑎𝑙 ≤ −𝑍𝛼
2
OR
𝑍𝑐𝑎𝑙 ≥ 𝑍𝛼
2
𝐻1: 𝜌12.3 > 𝜌0 𝑍𝑐𝑎𝑙𝑐 ≥ 𝑍𝛼
𝐻1: 𝜌12.3 < 𝜌0 𝑍𝑐𝑎𝑙𝑐 ≤ −𝑍𝛼

19. Example
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• Find simple correlation coefficients r12, r23, r13 and
interpret the results
• Calculate partial correlation coefficients r13.2, r12.3, r23.1
and interpret the results, Also test the hypothesis that
ρ12.3= 0.70
Results:
n= 6, r12= -0.891, r13= -0.969, r23= 0.961
X1 X2 X3
3 16 90
5 10 72
6 7 54
8 4 42
12 3 30
14 2 12

20. Multiple Correlation
• Multiple correlation is a measure of the linear relationship between a single
dependent variable and a set of explanatory variables
• If X1, X2, and X3 are three variables and we want to measure the combined
effect of X2 and X3 on X1, then the Population correlation coefficient is denoted
by ρ1.23 (read as rho one dot two three) and can be calculated as
• Its value is always between zero and 1. The R2
1.23 is the same quantity as is the
coefficient of multiple determination, calculated in a multiple regression taking
X1 response and X2, X3 as explanatory variables.
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𝑅1.23 =
𝑟12
2
+ 𝑟13
2
− 2𝑟12𝑟13𝑟23
1 − 𝑟23
2

21. R 2.13 and R 3.12
• Find multiple correlation coefficients R1.32, R2.13 and R3.12 and interpret the
results. Test the hypothesis that ρ3.12= 0
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𝑅2.13 =
𝑟12
2
+ 𝑟23
2
− 2𝑟12𝑟13𝑟23
1 − 𝑟13
2
𝑅3.12 =
𝑟13
2
+ 𝑟23
2
− 2𝑟12𝑟13𝑟23
1 − 𝑟12
2

22. Testing of hypothesis for Multiple Correlation
• Step-I: Formulation of hypothesis
• Step-II: Level of Significance: α=0.05
• Step-III: Test Statistics
• Step-IV: Calculations:
• Step-V: Decision Rule:
• Step-VI: Conclusion
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To test whether the multiple correlation coefficient ρ12.3 is equal to ZERO or not
𝐻0: 𝜌1.23 = 0
𝐻1: 𝜌1.23 ≠ 0
𝐹 =
𝑛 − 𝑞 − 1 𝑅1.23
2
𝑞 1 − 𝑅1.23
2
where q is the no. of variables whose combined effect is being seen on a response variable i. e. , in 𝑅1.23, q = 2
F𝐶𝑎𝑙𝑐 > 𝐹𝛼;(𝑞 ,𝑛 − 𝑞 − 1)

23. Example
The marks in Statistics (X1) are expressed as a function of marks in
Mathematics (X2), Economics (X3) and intelligence tests (X4). For a random
sample of 50 students, the Multiple Correlation Co-efficient R1.234 was found
to be 0.582. Test the hypothesis that the Multiple Correlation Co-efficient in
the Population is zero at α=0.05.
Solution:
Fcalc=7.87
Fα(q, n-q-1)=F0.05(3, 46)=2.81
Conclusion: Since the calculated value of F falls in the critical region, so we
reject the Null Hypothesis and may conclude that the Multiple Correlation
Co-efficient in the Population differs from zero Significantly.
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